# A property of span of S

As we know, ${S}$ is a nonempty subset of a vector space ${V}$, then the set ${W}$ consisting of all linear combinations of elements of ${S}$ is a subspace of ${V}$.

The subspace ${W}$ described in above fact is called the span of ${S}$ {or the subspace generated by the elements of ${S}$).

We have the following theorem in the book ‘Linear algebra’ of Friedberg et al.

Theorem 1 Let ${S}$ be a linearly independent subset of a vector space ${V}$, and let ${v}$ be an element of ${V}$ that is not in ${S}$. Then ${S \cup \{v\}}$ is linearly dependent if and only if ${ v \in span(S)}$.